Answer :
In order to determine the possible values for \( h \) given a triangle with side lengths \( 3x \) cm, \( 7x \) cm, and \( h \) cm, we need to use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Thus, for our triangle:
1. \( 3x + 7x > h \)
2. \( 3x + h > 7x \)
3. \( 7x + h > 3x \)
Let's solve these inequalities one by one:
1. \( 3x + 7x > h \)
[tex]\[ 10x > h \][/tex]
[tex]\[ h < 10x \][/tex]
2. \( 3x + h > 7x \)
[tex]\[ h > 7x - 3x \][/tex]
[tex]\[ h > 4x \][/tex]
3. \( 7x + h > 3x \)
[tex]\[ 7x + h > 3x \][/tex]
[tex]\[ h > 3x - 7x \][/tex]
[tex]\[ h > -4x \][/tex]
However, the inequality \( h > -4x \) is redundant because \( h > 4x \) is already a stricter condition given that \( x \) is positive.
Combining the first two relevant inequalities, we get:
[tex]\[ 4x < h < 10x \][/tex]
Thus, the expression that describes the possible values of \( h \) is:
[tex]\[ 4x < h < 10x \][/tex]
So, the correct answer is:
[tex]\[ \boxed{4x < h < 10x} \][/tex]
Thus, for our triangle:
1. \( 3x + 7x > h \)
2. \( 3x + h > 7x \)
3. \( 7x + h > 3x \)
Let's solve these inequalities one by one:
1. \( 3x + 7x > h \)
[tex]\[ 10x > h \][/tex]
[tex]\[ h < 10x \][/tex]
2. \( 3x + h > 7x \)
[tex]\[ h > 7x - 3x \][/tex]
[tex]\[ h > 4x \][/tex]
3. \( 7x + h > 3x \)
[tex]\[ 7x + h > 3x \][/tex]
[tex]\[ h > 3x - 7x \][/tex]
[tex]\[ h > -4x \][/tex]
However, the inequality \( h > -4x \) is redundant because \( h > 4x \) is already a stricter condition given that \( x \) is positive.
Combining the first two relevant inequalities, we get:
[tex]\[ 4x < h < 10x \][/tex]
Thus, the expression that describes the possible values of \( h \) is:
[tex]\[ 4x < h < 10x \][/tex]
So, the correct answer is:
[tex]\[ \boxed{4x < h < 10x} \][/tex]
